Question

Solve this please.

• Log 2 + log(3/2) + log (4/3) + log (5/4)+ log(6/5) + log (7/6) + log (8/7)​

Answer 1

`3 \log 2`

Explanation:

Given expression:

`\log 2 + \log \left((3)/(2)\right)+\log \left((4)/(3)\right)+\log \left((5)/(4)\right)+\log \left((6)/(5)\right)+\log \left((7)/(6)\right)+\log \left((8)/(7)\right)`

`\textsf{Apply the \underline{Log Product Law}}: \quad \log_ax + \log_ay=\log_axy`

`\implies \log \left(2 \cdot (3)/(2) \cdot (4)/(3) \cdot (5)/(4) \cdot (6)/(5) \cdot (7)/(6) \cdot (8)/(7)\right)`

Cross out common factors:

`\implies \log \left(\diagup\!\!\!\!2 \cdot (\diagup\!\!\!\!3)/(\diagup\!\!\!\!2) \cdot (\diagup\!\!\!\!4)/(\diagup\!\!\!\!3) \cdot (\diagup\!\!\!\!5)/(\diagup\!\!\!\!4) \cdot (\diagup\!\!\!\!6)/(\diagup\!\!\!\!5) \cdot (\diagup\!\!\!\!7)/(\diagup\!\!\!\!6) \cdot (8)/(\diagup\!\!\!\!7)\right)`

Therefore:

`\implies \log 8`

Factor the number: 8 = 2³

`\implies \log 2^3`

`\textsf{Apply the \underline{Log Power Law}}: \quad \log_ax^n=n\log_ax`

`\implies 3 \log 2`